This paper is concerned with the plastic buckling of Mindlin plates of polygonal plan shape and whose straight edges are simply supported. The plates are subjected to a uniform in-plane compressive stress. Two well-known competing theories of plasticity are considered here: the incremental theory of plasticity (with the Prandtl-Reuss constitutive relations) and the deformation theory of plasticity (with the Hencky constitutive relation). Based on an analogy approach, the plastic buckling stresses of such Mindlin plates are expressed in terms of their corresponding elastic buckling stresses of Kirchhoff (classical thin) plates, albeit in a transcendental form. Using this buckling stress relationship and the readily available elastic buckling solutions, one may deduce the plastic buckling stresses of the corresponding Mindlin plates. Tabulated herein are some buckling stress factors for various polygonal shaped plates with material properties defined by the Ramberg-Osgood relation.